A varying magnetic field can accelerate charge particles, but it is said that a magnetic field can't do any work so it should not be able to speed up charged particles, right? How is this apparent contradiction resolved?
Answer
A varying magnetic field generates an electric field, and an electric field can do work on a particle. This is called Faraday's law of induction:
∇×→E=−∂→B∂t
The full Lorentz force equation is
→F=q(→E+→v×→B)
So for example, if the magnetic field is increasing in the ˆz direction, such that
→B=btˆz
and
∂→B∂t=bˆz
then the electric field is determined by
∇×→E=−bˆz
Thus the electric field is not zero, so work can be done on a charged particle as a result of a changing magnetic field.
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